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Chapter 0: Problem 19

Find the slope and \(y\) -intercept. $$y-3 x=6$$

### Short Answer

Expert verified

The slope is 3 and the y-intercept is 6.

## Step by step solution

01

## Rewrite the Equation in Slope-Intercept Form

The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept. Rewrite the given equation y - 3x = 6 in this form.

02

## Solve for y

Add 3x to both sides of the equation to get y = 3x + 6. Now the equation is in slope-intercept form.

03

## Identify the Slope

Compare the equation y = 3x + 6 to the general form y = mx + b. The coefficient of x is the slope, so m = 3.

04

## Identify the y-intercept

Compare the equation y = 3x + 6 to the general form y = mx + b. The constant term is the y-intercept, so b = 6.

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### linear equations

In mathematics, linear equations are equations of first degree. This means the highest exponent of the variable (commonly \(x\)) is 1. A linear equation generally looks like \(ax + b = 0\). In geometry, linear equations represent straight-line graphs.

For example, in the given equation \(y - 3x = 6\), it is linear because both \(y\) and \(x\) are to the first power. Converting it into \(y = 3x + 6\) doesn't change its nature; it still represents a straight line. Linear equations have many applications in different fields, including physics, economics, and biology.

###### slope

The slope, often denoted as \(m\), measures the steepness of a line. It is calculated as the 'rise' (the change in \(y\)) over the 'run' (the change in \(x\)). In the slope-intercept form of a linear equation, \(y = mx + b\), \(m\) is the slope.

For example, in the equation \(y = 3x + 6\), the coefficient of \(x\) is the slope. Therefore, the slope is \(m=3\). This means for every one unit increase in \(x\), \(y\) increases by 3 units. Knowing the slope helps in understanding how quickly or slowly the value of \(y\) changes with \(x\).

###### y-intercept

The \(y\)-intercept is where the line crosses the \(y\)-axis. In the slope-intercept form, \(y = mx + b\), \(b\) is the \(y\)-intercept.

For instance, in \(y = 3x + 6\), the constant term 6 is the \(y\)-intercept. This tells us that if \(x=0\), then \(y=6\). Therefore, the graph intersects the \(y\)-axis at the point (0, 6). The \(y\)-intercept is essential for graphing because it provides an initial point for drawing the line.

###### solving equations

Solving linear equations involves finding the values of variables that make the equation true. This often includes isolating the variable on one side of the equation.

For the equation \(y - 3x = 6\), adding \(3x\) to both sides rewrites it as \(y = 3x + 6\). This equation is now arranged in the slope-intercept form, making it easier to identify the slope and \(y\)-intercept.

Solving equations may involve other techniques like factoring or using the quadratic formula for different types of equations, but for linear equations, the steps are usually straightforward and involve basic algebraic operations.

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